Advanced Number Theory

          Module 1: Foundations of Number Theory

Section 1: Understanding the Number System
- Lesson 1: What Are Numbers?
  - Module 1, Section 1, Lesson 1 What Are Numbers?
- Lesson 2: Classifying Numbers: Natural, Integer, and Rational
  - Module 1, Section 1, Lesson 2 Classifying Numbers: Natural, Integer, and Rational

Section 2: Core Properties of Integers
- Lesson 1: Even and Odd Numbers: Recognizing Differences
  - Module 1, Section 2, Lesson 1 Even and Odd Numbers: Recognizing Differences
- Lesson 2: The Role of Zero and One in Number Theory
  - Module 1, Section 2, Lesson 2 The Role of Zero and One in Number Theory

Section 3: Basic Arithmetic Operations
- Lesson 1: Fundamentals of Addition and Subtraction
  - Module 1, Section 3, Lesson 1 Fundamentals of Addition and Subtraction
- Lesson 2: Introduction to Multiplication and Division
  - Module 1, Section 3, Lesson 2 Introduction to Multiplication and Division

Section 4: The Number Line and Magnitude
- Lesson 1: Understanding the Number Line
  - Module 1, Section 4, Lesson 1 Understanding the Number Line
- Lesson 2: Visualizing Size and Order of Numbers
  - Module 1, Section 4, Lesson 2 Visualizing Size and Order of Numbers

Section 5: Mathematical Notation and Language
- Lesson 1: Symbols and Conventions in Number Theory
  - Module 1, Section 5, Lesson 1 Symbols and Conventions in Number Theory
- Lesson 2: Writing and Interpreting Mathematical Statements
  - Module 1, Section 5, Lesson 2 Writing and Interpreting Mathematical Statements


Module 2: Divisibility and Factorization Basics

Section 1: Introduction to Divisibility
- Lesson 1: What Is Divisibility?
  - Module 2, Section 1, Lesson 1 What Is Divisibility?
- Lesson 2: Simple Divisibility Rules and Their Uses
  - Module 2, Section 1, Lesson 2 Simple Divisibility Rules and Their Uses

Section 2: Factors and Multiples
- Lesson 1: Identifying Factors of a Number
  - Module 2, Section 2, Lesson 1 Identifying Factors of a Number
- Lesson 2: Understanding Multiples and Their Patterns
  - Module 2, Section 2, Lesson 2 Understanding Multiples and Their Patterns

Section 3: Fundamentals of Prime Numbers
- Lesson 1: What Makes a Number Prime?
  - Module 2, Section 3, Lesson 1 What Makes a Number Prime?
- Lesson 2: Recognizing Prime Numbers in Everyday Contexts
  - Module 2, Section 3, Lesson 2 Recognizing Prime Numbers in Everyday Contexts

Section 4: Factor Trees and Decomposition
- Lesson 1: Constructing Factor Trees
  - Module 2, Section 4, Lesson 1 Constructing Factor Trees
- Lesson 2: Exploring the Unique Factorization Principle
  - Module 2, Section 4, Lesson 2 Exploring the Unique Factorization Principle

Section 5: Practical Applications of Divisibility
- Lesson 1: Divisibility in Simple Problem Solving
  - Module 2, Section 5, Lesson 1 Divisibility in Simple Problem Solving
- Lesson 2: Puzzles and Games Using Divisibility
  - Module 2, Section 5, Lesson 2 Puzzles and Games Using Divisibility


Module 3: Prime Numbers and Their Properties

Section 1: In-Depth Look at Prime Numbers
- Lesson 1: Essential Properties of Prime Numbers
  - Module 3, Section 1, Lesson 1 Essential Properties of Prime Numbers
- Lesson 2: Why Prime Numbers Matter in Mathematics
  - Module 3, Section 1, Lesson 2 Why Prime Numbers Matter in Mathematics

Section 2: Methods of Primality Testing
- Lesson 1: Simple Methods for Testing Primality
  - Module 3, Section 2, Lesson 1 Simple Methods for Testing Primality
- Lesson 2: Introduction to the Sieve of Eratosthenes
  - Module 3, Section 2, Lesson 2 Introduction to the Sieve of Eratosthenes

Section 3: Techniques of Prime Factorization
- Lesson 1: Step-by-Step Prime Factorization
  - Module 3, Section 3, Lesson 1 Step-by-Step Prime Factorization
- Lesson 2: Verifying Factorizations and Their Uniqueness
  - Module 3, Section 3, Lesson 2 Verifying Factorizations and Their Uniqueness

Section 4: The Role of Primes in Equations
- Lesson 1: How Primes Influence Equation Solutions
  - Module 3, Section 4, Lesson 1 How Primes Influence Equation Solutions
- Lesson 2: Recognizing Patterns Involving Prime Numbers
  - Module 3, Section 4, Lesson 2 Recognizing Patterns Involving Prime Numbers

Section 5: Historical Perspectives on Prime Numbers
- Lesson 1: Ancient Views and Discoveries of Primes
  - Module 3, Section 5, Lesson 1 Ancient Views and Discoveries of Primes
- Lesson 2: Evolution of Prime Number Theory Over Time
  - Module 3, Section 5, Lesson 2 Evolution of Prime Number Theory Over Time


Module 4: Modular Arithmetic and Congruences

Section 1: Introduction to Modular Arithmetic
- Lesson 1: Understanding the Modulo Operation
  - Module 4, Section 1, Lesson 1 Understanding the Modulo Operation
- Lesson 2: Everyday Applications of Modular Thinking
  - Module 4, Section 1, Lesson 2 Everyday Applications of Modular Thinking

Section 2: Grasping Congruence Relations
- Lesson 1: Basics of Congruence in Number Theory
  - Module 4, Section 2, Lesson 1 Basics of Congruence in Number Theory
- Lesson 2: Properties and Simple Examples of Congruence
  - Module 4, Section 2, Lesson 2 Properties and Simple Examples of Congruence

Section 3: Exploring Remainders
- Lesson 1: Techniques for Calculating Remainders
  - Module 4, Section 3, Lesson 1 Techniques for Calculating Remainders
- Lesson 2: Patterns Observed in Remainder Calculations
  - Module 4, Section 3, Lesson 2 Patterns Observed in Remainder Calculations

Section 4: Solving Simple Modular Equations
- Lesson 1: Step-by-Step Methods for Modular Equations
  - Module 4, Section 4, Lesson 1 Step-by-Step Methods for Modular Equations
- Lesson 2: Practice Problems in Modular Arithmetic
  - Module 4, Section 4, Lesson 2 Practice Problems in Modular Arithmetic

Section 5: Applications of Modular Arithmetic
- Lesson 1: An Introduction to Modular Arithmetic in Cryptography
  - Module 4, Section 5, Lesson 1 An Introduction to Modular Arithmetic in Cryptography
- Lesson 2: Recognizing Cycles and Patterns in Modular Systems
  - Module 4, Section 5, Lesson 2 Recognizing Cycles and Patterns in Modular Systems


Module 5: Patterns, Logic, and Applications in Number Theory

Section 1: Recognizing Patterns in Numbers
- Lesson 1: Observing Patterns in Numerical Sequences
  - Module 5, Section 1, Lesson 1 Observing Patterns in Numerical Sequences
- Lesson 2: Developing Skills in Pattern Recognition
  - Module 5, Section 1, Lesson 2 Developing Skills in Pattern Recognition

Section 2: Introduction to Logical Reasoning
- Lesson 1: Basic Logical Reasoning in Mathematics
  - Module 5, Section 2, Lesson 1 Basic Logical Reasoning in Mathematics
- Lesson 2: Using Logic to Validate Number Theory Concepts
  - Module 5, Section 2, Lesson 2 Using Logic to Validate Number Theory Concepts

Section 3: Problem Solving Strategies in Number Theory
- Lesson 1: Approaches to Tackling Number Theory Problems
  - Module 5, Section 3, Lesson 1 Approaches to Tackling Number Theory Problems
- Lesson 2: Breaking Down Complex Problems into Steps
  - Module 5, Section 3, Lesson 2 Breaking Down Complex Problems into Steps

Section 4: Applying Fundamental Theorems
- Lesson 1: Introduction to the Fundamental Theorem of Arithmetic
  - Module 5, Section 4, Lesson 1 Introduction to the Fundamental Theorem of Arithmetic
- Lesson 2: Practical Applications of Fundamental Concepts
  - Module 5, Section 4, Lesson 2 Practical Applications of Fundamental Concepts

Section 5: Review and Creative Challenges
- Lesson 1: Recap of Essential Number Theory Ideas
  - Module 5, Section 5, Lesson 1 Recap of Essential Number Theory Ideas
- Lesson 2: Creative Challenges and Exploration in Number Theory
  - Module 5, Section 5, Lesson 2 Creative Challenges and Exploration in Number Theory